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GALOIS FUNCTORS AND ENTWINING STRUCTURES
, 909
"... Abstract. Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injecti ..."
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Abstract. Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a grouplike natural transformation g: I → G generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to comonoidal functors on monoidal categories in the sense of A. Bruguières and A. Virelizier. As wellknow, for any set G the product G × − defines an endofunctor on the category of sets and this is a Hopf monad if and only if G allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to Galois objects in the sense of Chase and Sweedler.
BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
BIMONADS AND HOPF MONADS ON CATEGORIES BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
BIMONADICITY AND THE EXPLICIT BASIS PROPERTY
"... Abstract. Let L ⊣ R: X → Y be an adjunction with R monadic and L comonadic. Denote the induced monad on Y by M and the induced comonad on X by C. We characterize those C such that M satisfies the Explicit Basis property. We also discuss some new examples and results motivated by this characterizatio ..."
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Abstract. Let L ⊣ R: X → Y be an adjunction with R monadic and L comonadic. Denote the induced monad on Y by M and the induced comonad on X by C. We characterize those C such that M satisfies the Explicit Basis property. We also discuss some new examples and results motivated by this characterization. 1. The Explicit Basis and Redundant Coassociativity properties In May 2010 Lawvere conjectured that the unit law implies the associative law for comonads arising from EB monads as defined in [14]. The present paper grew out of the intention to understand that conjecture. Let C = (C, ε, δ) be a comonad on a category X. 1.1. Definition. A precoalgebra is a pair (X, s) where s: X → CX is a map in X such that the diagram below X s �� CX commutes. id (Of course, precoalgebras are just ‘coalgebras for the copointed endofunctor (C, ε)’; but we will need to consider both coalgebras and precoalgebras for the comonad C and, for this, it is more efficient to have a different name.) Now fix an adjunction L ⊣ R: X → Y with unit η: Id → RL and counit ε: LR → Id. Let C = LR: X → X and denote the induced comonad on X by C = (C, ε, δ). Every precoalgebra (X, s) induces a coreflexive pair RX ηR
NOTES ON BIMONADS AND HOPF MONADS
"... Abstract. For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a preHopf monad by requiring on ..."
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Abstract. For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a preHopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the preHopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered. 1.
3. Actions on functors and Galois fun...
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
QF FUNCTORS AND (CO)MONADS
"... Abstract. One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K: A → B between categories is Frobenius if there exists a functor G: B → A which is at the same time a right and left adjoint of K; a monad F on ..."
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Abstract. One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K: A → B between categories is Frobenius if there exists a functor G: B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the category of Fmodules. With these notions, an algebra A over a field k is a Frobenius algebra if and only if A ⊗k − is a Frobenius monad on the category of kvector spaces. The purpose of this paper is to find characterisations of quasiFrobenius algebras by just referring to constructions available in any categories. To achieve this we define QF functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors. QF monads on a category A are those monads F for which the forgetful functor UF: AF → A is a QF functor. Applied to module categories (or Grothendieck categories), our notions coincide with definitions first given K. Morita (and others). Further applications